Asymptotics of Dirichlet eigenvalues and eigenfunctions of the Laplacian on thin domains in R^d

By Denis I. Borisov, Pedro J. Freitas. 2009

75 citations recorded by the source corpus.

We consider the Laplace operator with Dirichlet boundary conditions on a domain in R^d and study the effect that performing a scaling in one direction has on the eigenvalues and corresponding eigenfunctions as a function of the scaling parameter around zero. This generalizes our previous results in two dimensions and, as in that case, allows us to obtain an approximation for Dirichlet eigenvalues for a large class of domains, under very mild assumptions. As an application, we derive a three--term asymptotic expansion for the first eigenvalue of d-dimensional ellipsoids.

Subfields: Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c14ec99c81d131a0733cc0ebcca36e0d294cf3b6